Chapter 4. Continuous Random Variables

Transcription

1 Chapter 4. Continuous Random Variables

2 Review Continuous random variable: A random variable that can take any value on an interval of R. Distribution: A density function f : R R + such that 1. non-negative, i.e., f(x) 0 for all x. 2. for every subset I R, 3. P (X I) = R I f(x) dx f(x) dx = 1.

3 Graphic description: (1) density f, (2) cdf F. Relation between density f and cdf F : F (x) = P (X x) = f(x) = F (x). x f(y) dy Remark: Given a continuous random variable X, for any real number a. Therefore, P (X = a) = 0 P (X < a) = P (X a), P (X > a) = P (X a) P (a < X < b) = P (a X < b) = P (a < X b) = P (a X b).

4 Example: Suppose X has density f that takes the following form: { cx(2 x), 0 x 2, f(x) = 0, otherwise. Determine c, cdf F, and find P (X < 1).

5 Expected Value, Variance, and Standard Deviation Continuous random variable X with density f. Expectation (expected value, mean, µ ): EX. = xf(x) dx

6 Theorem: Consider a function h and random variable h(x). Then E[h(X)] = h(x)f(x) dx Corollary: Let a, b be real numbers. Then E[aX + b] = aex + b. Corollary: Consider functions h 1,..., h k. Then E[h 1 (X) + + h k (X)] = E[h 1 (X)] + + E[h k (X)].

7 Variance ( σ 2 ) and Standard deviation ( σ ):. Var[X] = E [ (X EX) 2]., Std[X] = VarX. Proposition: Var[X] = 0 if and only if P (X = c) = 1 for some constant c. Proposition: Var[X] = E[X 2 ] (EX) 2. Proposition: Let a, b be real numbers. Then Var[aX + b] = a 2 Var[X].

8 Uniform Distribution Uniform distribution on [0, 1]. A random variable X with density { 1, 0 x 1 f(x) = 0, otherwise. Expected value and variance.

9 Uniform distribution on [a, b]. A random variable X with density 1 f(x) = b a, a x b 0, otherwise. Remark: If X is uniformly distributed on [0, 1], then Y. = (b a)x + a is uniformly distributed on [a, b]. Expected value and variance.

10 Example 1.(a) A point is randomly chosen from interval [ 1, 1]. Find the probability that the point is non-negative. (b) 1000 points are randomly chosen from interval [ 1, 1]. What do you think their average should be close to?

11 2. Suppose customers arrive at post office randomly such that The number of arrivals on any time interval with length s is Poisson distributed with parameter λs. The numbers of arrivals on disjoint time intervals are independent. Define event Let A. = {on time interval [0, T ], there is in total one customer}. X. = the arrival time of the first customer. (a) Compute P (X t A) for all t. (b) What is the distribution of X given that event A happens?

12 Exponential Distribution Exponential distribution with rate λ. A random variable X with density { λe λx, 0 x < f(x) = 0, otherwise. EX = 1 λ, Var[X] = 1 λ 2. Remark: Exponential distribution with rate λ is called in the textbook Exponential distribution with parameter β where β = 1/λ.

13 Example 1. Suppose X is exponentially distributed with rate λ. Find P (X > t).

14 2. Suppose customers arrive at post office randomly such that The number of arrivals on any time interval with length s is Poisson distributed with parameter λs. The numbers of arrivals on disjoint time intervals are independent. Let X =. the arrival time of the first customer. Compute P (X t). What is the distribution of X?

15 Memoryless property Suppose X is exponentially distributed with rate λ. Then P (X > t + s X > t) = P (X > s). Comment: Exponential distribution is the only continuous distribution that has memoryless property. Example: The life length of a computer is exponentially distributed with mean 5 years. You bought an old (working) computer for 10 dollars. What is the probability that it would still work for more than 3 years?

16 Normal Distribution Normal distribution N(µ, σ 2 ). A random variable X is normally distributed with mean µ and variance σ 2 if it has density f(x) = 1 } { σ 2π exp (x µ)2, x R. 2σ 2 1. f defines a probability density function. 2. EX = µ. 3. Var[X] = σ 2.

17 Standard Normal Distribution The standard normal means N(0, 1). Its cdf is often denoted by Φ, that is, x } 1 Φ(x) = exp { y2 dy 2π 2 Suppose Z is N(0, 1), then In particular, P (Z > x) = P (Z < x) = Φ( x). Φ(x) + Φ( x) = 1.

18 Standardization Theorem: Suppose X has normal distribution N(µ, σ 2 ). Then its linear transform Y. = ax +b is also normally distributed with distribution N(aµ+b, a 2 σ 2 ). Corollary: Suppose X has distribution N(µ, σ 2 ). Then its standardization Z =. X µ σ is standard normal.

19 Examples 1. Suppose X has distribution N(40, 100). Find P (50 < X < 60). 2. Suppose X has distribution N(µ, σ 2 ). Then In particular, P ( X µ > kσ) = 2Φ( k). k Φ( k) 32% 5% 0.3% Remark: Normal distribution is often used as approximation of non-negative distributions. For example, SAT scores, IQ, GRE scores